Jet schemes and invariant theory

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin

doi:10.5802/aif.2996

Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin

Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599.

Abstract
Résumé

Let $G$ be a complex reductive group and $V$ a $G$ -module. Then the $m$ th jet scheme $G_{m}$ acts on the $m$ th jet scheme $V_{m}$ for all $m \geq 0$ . We are interested in the invariant ring $𝒪 {(V_{m})}^{G_{m}}$ and whether the map $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induced by the categorical quotient map $p : V \to V / / G$ is an isomorphism, surjective, or neither. Using Luna’s slice theorem, we give criteria for $p_{m}^{*}$ to be an isomorphism for all $m$ , and we prove this when $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , or ${Sp}_{2 n}$ and $V$ is a sum of copies of the standard module and its dual, such that $V / / G$ is smooth or a complete intersection. We classify all representations of $ℂ^{*}$ for which $p_{\infty}^{*}$ is surjective or an isomorphism. Finally, we give examples where $p_{m}^{*}$ is surjective for $m = \infty$ but not for finite $m$ , and where it is surjective but not injective.

Soient $G$ un groupe réductif complexe et $V$ un $G$ -module. Alors $G_{m}$ , le schéma des jets d’ordre $m$ de $G$ , opère dans $V_{m}$ , le schéma des jets d’ordre $m$ de $V$ , pour tout $m \geq 0$ . Nous nous intéressons à l’anneau des invariants $𝒪 {(V_{m})}^{G_{m}}$ et au morphisme $p_{m}^{*} : 𝒪 ({(V / / G)}_{m}) \to 𝒪 {(V_{m})}^{G_{m}}$ induit par le morphisme du quotient catégorique $p : V \to V / / G$ : ce morphisme est-il un isomorphisme, surjectif, ou non ? En utilisant le théorème du slice de Luna, nous obtenons des critères pour que $p_{m}^{*}$ soit un isomorphisme pour tout $m$ . Nous montrons que c’est bien le cas lorsque $G = {SL}_{n}$ , ${GL}_{n}$ , ${SO}_{n}$ , ou ${Sp}_{2 n}$ et $V$ est un somme directe de copies du module standard et de son dual, pourvu que $V / / G$ soit lisse ou une intersection complète. Nous classifions toutes les représentations de $ℂ^{*}$ telles que $p_{\infty}^{*}$ soit surjectif ou un isomorphisme. Enfin, nous donnons des exemples où $p_{m}^{*}$ est surjectif pour $m = \infty$ mais non surjectif pour $m$ fini, et d’autres exemples où $p_{m}^{*}$ est surjectif mais non injectif.

Received: 2014-10-03
Accepted: 2015-02-24
Published online: 2015-12-08

Zbl: 1342.13009

DOI: 10.5802/aif.2996
Classification: 13A50, 14L24, 14L30
Keywords: jet schemes, classical invariant theory

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     title = {Jet schemes and invariant theory},
     journal = {Annales de l'Institut Fourier},
     pages = {2571--2599},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2996},
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     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2996/}
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Linshaw, Andrew R.; Schwarz, Gerald W.; Song, Bailin. Jet schemes and invariant theory. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2571-2599. doi : 10.5802/aif.2996. https://aif.centre-mersenne.org/articles/10.5802/aif.2996/

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